Cox Proportional Hazards Model

Hermela Shimelis and Kayla Boyd

November 18, 2024

Outline

  • Introduction of Cox regression

  • Two examples

    • Survival after chemotherapy of colon cancer patients
    • Kayla’s data

Cox proportional hazards model

  • A popular regression modeling method used to explore the relationship between survival time and covariates.


  • It assumes that the effects of different variables on the outcome are constant over time.


  • Survival can refer to the development of a symptom, time to relapse after remission, or as a time to death [1].


Cox proportional hazards model

  • Cox regression model is based on the hazard function \(h_x(t)\) with covariates at time t given by [2]:

    • \(h_x(t)=h_0(t)\exp(\beta_1x_1 +\beta_2x_2 + \dots + \beta_p x_p)\)

    • Where:

      • \(h_x(t)\) is the hazard function

      • \(h_0(t)\) is the baseline hazard function

      • \(\beta_1x_1 + \beta_2x_2 + \dots +\beta_p x_p\) represent the linear combination of covariates and their coefficient

Proportional Hazards assumption

  • The assumption of a constant relationship between dependent and explanatory variables is called proportional hazards [3].

Hazard Ratios

  • The hazard function is the probability that an individual will experience an event (death) within a certain time interval [1].

  • The hazard ratio is used to compare the hazard rate between two groups

    • HR = \(hx_2(t)\) / \(hx_1(t)\) = \(\exp[\beta(x_2-x_1)]\)

      • where \(hx_2(t)\) and \(hx_1(t)\) are the hazard function for the two group

    • HR = 1: No difference in hazard rates between the two groups

    • HR >1: Higher hazard rate in the second group compared to the first

    • HR <1: Lower hazard rate in the second group compared to the first

Time-Varying Coefficients

  • Failing to meet the assumption of proportional hazards means that the effects between dependent and explanatory variables are not constant over time.

  • Time-varying covariates (coefficients) are used when a covariate changes over time during the follow-up period [4].

  • Internal time-varying coefficients are affected by survival status and include values that are generated by the subject [4].

  • External time-varying coefficients are pre-determined and external to the subject under study [4].

R packages used for survival analysis

Package Description
Survival - Used for fitting and analyzing survival models
- Fits Kaplan-Meier survival curves
Survminer - Plots Kaplan-Meier survival curves using ggplot2
- Plots Schoenfeld residuals

Cox regression modeling of survival after chemotherapy for colon cancer

  • Data: Survival after chemotherapy for Stage B/C colon cancer [6]

  • Goal: Model the relationship between survival time and treatment groups

  • Predictors

Category Variables
Treatments - Observation (no treatment)
- Amisole (Lev)
- Amisole + 5-FU
Patient Characteristics - Age
- Sex
Tumor Characteristics - Colon perforation and obstruction
- Adherence to nearby organs
- Tumor differentiation
- Local spread

Kaplan-Meier curve stratified by treatment groups

Cox regression models

  1. Base model: No predictors

  2. Univariate: Treatment

  3. Full variables: All predictors

  4. Significant predictors: stepwise-selected variables

  5. Final model: Stratified

Model 2: Univariate model

Characteristic HR1 95% CI1 p-value
rx


    Obs — —
    Lev 0.97 0.78, 1.21 0.8
    Lev+5FU 0.69 0.55, 0.87 0.002
Concordance = 0.536
1 HR = Hazard Ratio, CI = Confidence Interval

Model 3: All predictors

Characteristic HR1 95% CI1 p-value
rx


    Obs — —
    Lev 0.98 0.79, 1.22 0.9
    Lev+5FU 0.69 0.54, 0.87 0.002
age 1.01 1.00, 1.02 0.083
sex 1.04 0.86, 1.26 0.7
perfor 1.00 0.59, 1.70 >0.9
adhere 1.18 0.92, 1.53 0.2
surg 1.27 1.03, 1.55 0.022
obstruct 1.33 1.06, 1.68 0.015
differentiation


    moderate — —
    poor 1.43 1.13, 1.82 0.003
    well 1.08 0.78, 1.50 0.6
node4 2.55 2.10, 3.09 <0.001
local_spread


    contiguous — —
    muscle 0.39 0.23, 0.64 <0.001
    serosa 0.64 0.43, 0.94 0.023
    submucosa 0.29 0.10, 0.83 0.021
Concordance = 0.674
1 HR = Hazard Ratio, CI = Confidence Interval

Model 4: Stepwise-selected variables

Characteristic HR1 95% CI1 p-value
rx


    Obs — —
    Lev 0.99 0.80, 1.23 >0.9
    Lev+5FU 0.69 0.54, 0.87 0.002
age 1.01 1.00, 1.02 0.069
surg 1.28 1.04, 1.56 0.018
obstruct 1.33 1.06, 1.67 0.015
differentiation


    moderate — —
    poor 1.45 1.15, 1.84 0.002
    well 1.07 0.77, 1.48 0.7
node4 2.53 2.09, 3.07 <0.001
local_spread


    contiguous — —
    muscle 0.37 0.23, 0.61 <0.001
    serosa 0.61 0.41, 0.89 0.010
    submucosa 0.27 0.09, 0.76 0.014
Concordance = 0.672
1 HR = Hazard Ratio, CI = Confidence Interval

Proportional Hazard Assumption is not met

Schoenfeld Residuals Test Results
chisq df p Variable
rx 2.335 2 0.311 rx
age 0.549 1 0.459 age
surg 0.020 1 0.888 surg
obstruct 6.148 1 0.013 obstruct
differentiation 17.459 2 0.000 differentiation
node4 5.662 1 0.017 node4
local_spread 7.083 3 0.069 local_spread
GLOBAL 37.525 11 0.000 GLOBAL

Model 5: Stratified Model

Characteristic HR1 95% CI1 p-value
rx


    Obs — —
    Lev 0.98 0.79, 1.22 0.9
    Lev+5FU 0.71 0.56, 0.89 0.003
age 1.01 1.00, 1.02 0.034
surg 1.30 1.06, 1.59 0.012
node4 2.50 2.06, 3.04 <0.001
local_spread


    contiguous — —
    muscle 0.34 0.21, 0.56 <0.001
    serosa 0.58 0.39, 0.84 0.004
    submucosa 0.24 0.08, 0.67 0.007
Concordance = 0.674
1 HR = Hazard Ratio, CI = Confidence Interval

Stratified model meets proportional hazards assumption

Schoenfeld Residuals Test Results
chisq df p Variable
rx 2.001 2 0.368 rx
age 0.670 1 0.413 age
surg 0.014 1 0.905 surg
node4 4.288 1 0.038 node4
local_spread 5.298 3 0.151 local_spread
GLOBAL 12.411 8 0.134 GLOBAL

Model Evaluation Metrics

Model Description AIC BIC C_Index
Model 1 Base model 5860.383 5860.383 0.500
Model 2 Treatment 5852.236 5860.463 0.536
Model 3 Full variables 5741.401 5798.993 0.674
Model 4 Stepwise-selected variables 5737.261 5782.511 0.672
Model 5 Stratified 4567.829 4600.739 0.674

K-fold cross validation

Original c-index: 0.6544784 
Mean cross-validated c-Index: 0.6420104

Conclusions

References

[1]
S. J. Walters, “Analyzing time to event outcomes with a cox regression model,” Wiley Interdiscip. Rev. Comput. Stat., vol. 4, no. 3, pp. 310–315, May 2012.
[2]
R. Tibshirani, “What is cox’s proportional hazards model?” Signif. (Oxf.), vol. 19, no. 2, pp. 38–39, Apr. 2022.
[3]
C. A. Bellera, G. MacGrogan, M. Debled, C. T. de Lara, V. Brouste, and S. Mathoulin-Pélissier, “Variables with time-varying effects and the cox model: Some statistical concepts illustrated with a prognostic factor study in breast cancer,” BMC Med. Res. Methodol., vol. 10, no. 1, p. 20, Mar. 2010.
[4]
Z. Zhang, J. Reinikainen, K. A. Adeleke, M. E. Pieterse, and C. G. M. Groothuis-Oudshoorn, “Time-varying covariates and coefficients in cox regression models,” Ann. Transl. Med., vol. 6, no. 7, pp. 121–121, Apr. 2018.
[5]
Terry M. Therneau and Patricia M. Grambsch, Modeling survival data: Extending the Cox model. New York: Springer, 2000.
[6]
T. M. Therneau, A package for survival analysis in r. 2024. Available: https://CRAN.R-project.org/package=survival